On a Nonstandard Boundary Value Problem Arising in Homogenization of Complex Heat Transfer Problems

Abstract

We consider a nonstandard boundary value problem appearing in homogenization of some radiative-conductive heat transfer problems. We establish the existence, uniqueness, and regularity of a weak solution u. We obtain estimates for the derivatives $$ {D}_1^2u,{D}_2^2u,\kern0.33em {D}_1{D}_2u\kern0.33em in\kern0.33em {L}^2\left({\varOmega}_{\upvarepsilon}\right)\kern0.33em and\kern0.33em {D}_s^2u\kern0.33em in\kern0.33em {L}^2\left({\varGamma}_{\varepsilon}\right) $$ with qualified order with respect to e, where Ωe is the square (e/2, 1 − e/2) × (e/2, 1 − e/2) with boundary Γe.